For our model, the tree-level vacuum stability is implemented in Eq. (REF ). For the vacuum stability
at one-loop level, we follow the gauge-invariant treatment used in Ref. [1]} by requiring
\(\lambda _{h}(\mu )>0,~~ \lambda _{s}(\mu )>0,~~ \lambda _{h}(\mu ) \lambda _{s}(\mu )>\frac{1}{4} \lambda _{m}(\mu )^{2}, {~~\rm for~~} \mu <\Lambda ,\) 
